Bipartite and tripartite entanglement in pure dephasing relativistic spin-boson model

TL;DR

The paper investigates nonperturbative entanglement generation among two and three emitters within a solvable relativistic spin-boson framework that mirrors a time-independent Unruh–DeWitt model. By enforcing pure dephasing and carefully controlling UV/IR regularity, it reveals that significant bipartite entanglement requires interactions deep inside the light cone, with field mass generally enhancing entanglement at the cost of longer interaction times, and that genuine GHZ-like tripartite entanglement occurs only in a narrow parameter space. The results emphasize the limitations of the gapless model for robust multipartite entanglement studies and suggest that alternative models or probe configurations may be necessary to access richer multipartite quantum-field entanglement. Overall, the work provides rigorous ground-state regularity conditions and a clear causal-signal framework that clarifies the role of mass, dimensionality, and coupling in relativistic entanglement dynamics.

Abstract

We study nonperturbatively the entanglement generation between two and three emitters in an exactly solvable relativistic variant of the spin-boson model, equivalent to the time-independent formulation of the Unruh-DeWitt detector model. We show that (i) (highly) entangled states of the two emitters require interactions very deep into the light cone, (ii) the mass of the field can generically improve the entanglement generation, (iii) while it is possible to find regimes with genuine Greenberger-Horne-Zeilinger-like tripartite entanglement, it is difficult find regimes where tripartite entanglement can be easily shown to be significant or classified. Result (iii), in particular, suggests that probing the multipartite entanglement of a relativistic quantum field nonperturbatively requires either different probe-based techniques or variants of the Unruh-DeWitt model. Along the way, we provide the regularity conditions for the $N$-emitter model to have well-defined ground states in the Fock space.

Figures (5)

• Figure 1: Density plots [(a-i) and (b-i)] of negativity $\mathcal{N}$ between two inertial detectors weakly coupled to the field ($\widetilde{\lambda} =0.01$) with $\Delta_{\mathrm{A}}=\Delta_{\mathrm{B}}=0$ in $(3+1)$-dimensional Minkowski spacetime when the mass of the field is (a) $m \sigma=10^{-11}$ and (b) $m \sigma=0.1$. The green dots depict the region where $\mathcal{N}=0$, and the white dashed line represents the light cone from the edge of the detector (which starts from $L/\sigma\approx 3.5$ at $t=0$), whose COM is located at the origin of the diagrams. On the right-hand side [(a-ii) and (b-ii)], the time dependence of $\mathcal{N}$ at $L/\sigma=10$ is depicted.
• Figure 2: Density plots [(a-i) and (b-i)] of negativity $\mathcal{N}$ between two inertial detectors strongly coupled ($\widetilde{\lambda} =1$) to the field in $(3+1)$-dimensional Minkowski spacetime when the mass of the field is (a) $m \sigma=10^{-11}$ and (b) $m \sigma=0.1$. The green dots depict the region where $\mathcal{N}=0$, and the white dashed line represents the light cone from the edge of the detector (which starts from $L/\sigma\approx 3.5$ at $t=0$), whose COM is located at the origin of the diagrams. On the right-hand side [(a-ii) and (b-ii)], the time dependence of $\mathcal{N}$ at $L/\sigma=10$ is depicted.
• Figure 3: Density plots [(a-i), (b-i), and (c-i)] of negativity $\mathcal{N}$ between two inertial detectors weakly coupled ($\widetilde{\lambda}=0.01$) to the nearly massless field in various spacetime dimensions. Here, $\widetilde{\lambda}\equiv \lambda \sigma^{-(n-3)/2}$ for $(n+1)$-dimensional Minkowski spacetime. On the right panel [(a-ii), (b-ii), and (c-ii)], the time dependence of $\mathcal{N}$ at $L/\sigma=10$ with $m \sigma=10^{-11}$ (blue plots) and $m\sigma=0.1$ (orange plots) are depicted.
• Figure 4: Density plots of negativity $\mathcal{N}$ between two inertial detectors in $(3+1)$-dimensional Minkowski spacetime, showing how mass $m$ of the field contributes to $\mathcal{N}$. (a) and (b) correspond to the weak coupling $\widetilde{\lambda}=0.01$ and strong coupling $\widetilde{\lambda} =1$ cases. (a-i) and (b-i) depict the dependence on $m\sigma$ and $L/\sigma$ at a fixed time $t/\sigma=10^2$, whereas (a-ii) and (b-ii) show the dependence on $m\sigma$ and $t/\sigma$ at a fixed distance $L/\sigma=10$.
• Figure 5: Density plots of the positive $\pi$-tangle among three inertial detectors in the equilateral triangular configuration in $(3+1)$-dimensional Minkowski spacetime when the detectors are (a) weakly coupled, and (b) strongly coupled to the field. (a-i) and (b-i) show the nearly massless ($m \sigma =10^{-11}$) scenario, whereas (a-ii) and (b-ii) correspond to the massive case with $m \sigma=0.1$. The green dots depict the region where the $\pi$-tangle vanishes. The white dashed line represents the light cone from the edge of the detector as before, while the red dashed curve corresponds to the boundary of the bipartite entanglement cone depicted in Figs. \ref{['fig:4Ddensityweak']} and \ref{['fig:4DdensityStrong']}.

Theorems & Definitions (3)

• Proposition 1
• proof
• Proposition 2